Every healthcare system in the world including the socialized ones denies some patients life-saving care. Care will always be rationed regardless of whether some entities are making a profit.
These should be not-for-profit entities, with appropriate controls.
And private health insurers generally don't make more profit by denying claims. Due to the ACA minimum medical loss ratio rule it's rather the opposite. When claims are denied it's usually due to decisions made by self-funded employers that sponsor health plans for their employees. The insurers are mostly just acting as administrators for those plans rather than providing any real insurance.